Question

Write the equation of a parabola in vertex form with the vertex at (2, -8) and a directrix of x = 3.

Answer 1

For a parabola with vertex (h,k) = (2,-8) and horizontal axis, the vertex form is (y - k)^2 = 4p(x - h). The directrix x = 3 gives h - p = 3, so 2 - p = 3 ⇒ p = -1.

Thus
(y + 8)^2 = 4(-1)(x - 2) = -4(x - 2).

(Equivalently, x = 2 - 1/4 (y + 8)^2.)